LMFDB - Embedded newform 1470.3.f.d.391.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1470,3,Mod(391,1470)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1470, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1470.391");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1470.f (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$40.0545988610$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 92 x^{14} - 112 x^{13} + 5846 x^{12} - 7728 x^{11} + 197216 x^{10} - 298200 x^{9} + \cdots + 101626561 $ x^16 + 92x^14 - 112x^13 + 5846x^12 - 7728x^11 + 197216x^10 - 298200x^9 + 4836403x^8 - 6808704x^7 + 64376800x^6 - 91953512x^5 + 595763862x^4 - 630430976x^3 + 1087013404x^2 + 294123256x + 101626561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			391.3
Root			$-2.10711 - 3.64962i$ of defining polynomial
Character	$\chi$	$=$	1470.391
Dual form			1470.3.f.d.391.5

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} -3.16228i q^{10} -10.8220 q^{11} -3.46410i q^{12} +19.2715i q^{13} +3.87298 q^{15} +4.00000 q^{16} -10.2737i q^{17} +4.24264 q^{18} +20.8719i q^{19} +4.47214i q^{20} +15.3046 q^{22} +21.0746 q^{23} +4.89898i q^{24} -5.00000 q^{25} -27.2540i q^{26} +5.19615i q^{27} +19.0888 q^{29} -5.47723 q^{30} -40.0169i q^{31} -5.65685 q^{32} +18.7442i q^{33} +14.5292i q^{34} -6.00000 q^{36} -50.3654 q^{37} -29.5173i q^{38} +33.3792 q^{39} -6.32456i q^{40} -22.7706i q^{41} -48.4307 q^{43} -21.6440 q^{44} -6.70820i q^{45} -29.8040 q^{46} -66.5380i q^{47} -6.92820i q^{48} +7.07107 q^{50} -17.7946 q^{51} +38.5430i q^{52} +4.95062 q^{53} -7.34847i q^{54} -24.1987i q^{55} +36.1511 q^{57} -26.9957 q^{58} -28.1868i q^{59} +7.74597 q^{60} +70.0889i q^{61} +56.5924i q^{62} +8.00000 q^{64} -43.0923 q^{65} -26.5083i q^{66} +19.3057 q^{67} -20.5474i q^{68} -36.5023i q^{69} +49.4968 q^{71} +8.48528 q^{72} -132.974i q^{73} +71.2274 q^{74} +8.66025i q^{75} +41.7437i q^{76} -47.2053 q^{78} -90.0808 q^{79} +8.94427i q^{80} +9.00000 q^{81} +32.2026i q^{82} +101.045i q^{83} +22.9727 q^{85} +68.4913 q^{86} -33.0628i q^{87} +30.6092 q^{88} -39.6151i q^{89} +9.48683i q^{90} +42.1492 q^{92} -69.3113 q^{93} +94.0989i q^{94} -46.6709 q^{95} +9.79796i q^{96} -68.6944i q^{97} +32.4659 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) $ 16 * q + 32 * q^4 - 48 * q^9	$ 16 q + 32 q^{4} - 48 q^{9} + 8 q^{11} + 64 q^{16} - 48 q^{22} + 24 q^{23} - 80 q^{25} + 72 q^{29} - 96 q^{36} - 88 q^{37} - 72 q^{39} - 56 q^{43} + 16 q^{44} - 16 q^{46} + 24 q^{51} - 64 q^{53} + 144 q^{57} + 176 q^{58} + 128 q^{64} - 40 q^{65} + 328 q^{67} - 136 q^{71} + 224 q^{74} - 560 q^{79} + 144 q^{81} + 120 q^{85} + 176 q^{86} - 96 q^{88} + 48 q^{92} + 240 q^{93} - 400 q^{95} - 24 q^{99}+O(q^{100}) $ 16 * q + 32 * q^4 - 48 * q^9 + 8 * q^11 + 64 * q^16 - 48 * q^22 + 24 * q^23 - 80 * q^25 + 72 * q^29 - 96 * q^36 - 88 * q^37 - 72 * q^39 - 56 * q^43 + 16 * q^44 - 16 * q^46 + 24 * q^51 - 64 * q^53 + 144 * q^57 + 176 * q^58 + 128 * q^64 - 40 * q^65 + 328 * q^67 - 136 * q^71 + 224 * q^74 - 560 * q^79 + 144 * q^81 + 120 * q^85 + 176 * q^86 - 96 * q^88 + 48 * q^92 + 240 * q^93 - 400 * q^95 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times$.

$n$	$491$	$1081$	$1177$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.41421	−0.707107
$2$	−1.41421	−0.707107
$3$	− 1.73205i	− 0.577350i
$3$	− 1.73205i	− 0.577350i
$4$	2.00000	0.500000
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	2.44949i	0.408248i
$7$	0	0
$7$	0	0
$8$	−2.82843	−0.353553
$9$	−3.00000	−0.333333
$10$	− 3.16228i	− 0.316228i
$11$	−10.8220	−0.983816	−0.491908	−	0.870647i	$-0.663700\pi$
$11$	−10.8220	−0.983816	−0.491908	+	0.870647i	$0.663700\pi$
$12$	− 3.46410i	− 0.288675i
$13$	19.2715i	1.48242i	0.671273	+	0.741211i	$0.265748\pi$
$13$	19.2715i	1.48242i	−0.671273	+	0.741211i	$0.734252\pi$
$14$	0	0
$15$	3.87298	0.258199
$16$	4.00000	0.250000
$17$	− 10.2737i	− 0.604336i	−0.953255	−	0.302168i	$-0.902290\pi$
$17$	− 10.2737i	− 0.604336i	0.953255	−	0.302168i	$-0.0977104\pi$
$18$	4.24264	0.235702
$19$	20.8719i	1.09852i	0.835652	+	0.549260i	$0.185090\pi$
$19$	20.8719i	1.09852i	−0.835652	+	0.549260i	$0.814910\pi$
$20$	4.47214i	0.223607i
$21$	0	0
$22$	15.3046	0.695663
$23$	21.0746	0.916286	0.458143	−	0.888878i	$-0.348515\pi$
$23$	21.0746	0.916286	0.458143	+	0.888878i	$0.348515\pi$
$24$	4.89898i	0.204124i
$25$	−5.00000	−0.200000
$26$	− 27.2540i	− 1.04823i
$27$	5.19615i	0.192450i
$28$	0	0
$29$	19.0888	0.658235	0.329118	−	0.944289i	$-0.393249\pi$
$29$	19.0888	0.658235	0.329118	+	0.944289i	$0.393249\pi$
$30$	−5.47723	−0.182574
$31$	− 40.0169i	− 1.29087i	−0.763816	−	0.645434i	$-0.776677\pi$
$31$	− 40.0169i	− 1.29087i	0.763816	−	0.645434i	$-0.223323\pi$
$32$	−5.65685	−0.176777
$33$	18.7442i	0.568007i
$34$	14.5292i	0.427330i
$35$	0	0
$36$	−6.00000	−0.166667
$37$	−50.3654	−1.36123	−0.680614	−	0.732643i	$-0.738287\pi$
$37$	−50.3654	−1.36123	−0.680614	+	0.732643i	$0.738287\pi$
$38$	− 29.5173i	− 0.776771i
$39$	33.3792	0.855876
$40$	− 6.32456i	− 0.158114i
$41$	− 22.7706i	− 0.555382i	−0.960671	−	0.277691i	$-0.910431\pi$
$41$	− 22.7706i	− 0.555382i	0.960671	−	0.277691i	$-0.0895691\pi$
$42$	0	0
$43$	−48.4307	−1.12629	−0.563147	−	0.826357i	$-0.690410\pi$
$43$	−48.4307	−1.12629	−0.563147	+	0.826357i	$0.690410\pi$
$44$	−21.6440	−0.491908
$45$	− 6.70820i	− 0.149071i
$46$	−29.8040	−0.647912
$47$	− 66.5380i	− 1.41570i	−0.706362	−	0.707851i	$-0.749665\pi$
$47$	− 66.5380i	− 1.41570i	0.706362	−	0.707851i	$-0.250335\pi$
$48$	− 6.92820i	− 0.144338i
$49$	0	0
$50$	7.07107	0.141421
$51$	−17.7946	−0.348914
$52$	38.5430i	0.741211i
$53$	4.95062	0.0934079	0.0467040	−	0.998909i	$-0.485128\pi$
$53$	4.95062	0.0934079	0.0467040	+	0.998909i	$0.485128\pi$
$54$	− 7.34847i	− 0.136083i
$55$	− 24.1987i	− 0.439976i
$56$	0	0
$57$	36.1511	0.634231
$58$	−26.9957	−0.465442
$59$	− 28.1868i	− 0.477742i	−0.971051	−	0.238871i	$-0.923223\pi$
$59$	− 28.1868i	− 0.477742i	0.971051	−	0.238871i	$-0.0767773\pi$
$60$	7.74597	0.129099
$61$	70.0889i	1.14900i	0.818505	+	0.574499i	$0.194803\pi$
$61$	70.0889i	1.14900i	−0.818505	+	0.574499i	$0.805197\pi$
$62$	56.5924i	0.912781i
$63$	0	0
$64$	8.00000	0.125000
$65$	−43.0923	−0.662959
$66$	− 26.5083i	− 0.401641i
$67$	19.3057	0.288145	0.144073	−	0.989567i	$-0.453980\pi$
$67$	19.3057	0.288145	0.144073	+	0.989567i	$0.453980\pi$
$68$	− 20.5474i	− 0.302168i
$69$	− 36.5023i	− 0.529018i
$70$	0	0
$71$	49.4968	0.697138	0.348569	−	0.937283i	$-0.386668\pi$
$71$	49.4968	0.697138	0.348569	+	0.937283i	$0.386668\pi$
$72$	8.48528	0.117851
$73$	− 132.974i	− 1.82157i	−0.412886	−	0.910783i	$-0.635479\pi$
$73$	− 132.974i	− 1.82157i	0.412886	−	0.910783i	$-0.364521\pi$
$74$	71.2274	0.962533
$75$	8.66025i	0.115470i
$76$	41.7437i	0.549260i
$77$	0	0
$78$	−47.2053	−0.605196
$79$	−90.0808	−1.14026	−0.570132	−	0.821553i	$-0.693108\pi$
$79$	−90.0808	−1.14026	−0.570132	+	0.821553i	$0.693108\pi$
$80$	8.94427i	0.111803i
$81$	9.00000	0.111111
$82$	32.2026i	0.392714i
$83$	101.045i	1.21741i	0.793396	+	0.608706i	$0.208311\pi$
$83$	101.045i	1.21741i	−0.793396	+	0.608706i	$0.791689\pi$
$84$	0	0
$85$	22.9727	0.270267
$86$	68.4913	0.796410
$87$	− 33.0628i	− 0.380032i
$88$	30.6092	0.347832
$89$	− 39.6151i	− 0.445113i	−0.974920	−	0.222557i	$-0.928560\pi$
$89$	− 39.6151i	− 0.445113i	0.974920	−	0.222557i	$-0.0714402\pi$
$90$	9.48683i	0.105409i
$91$	0	0
$92$	42.1492	0.458143
$93$	−69.3113	−0.745283
$94$	94.0989i	1.00105i
$95$	−46.6709	−0.491273
$96$	9.79796i	0.102062i
$97$	− 68.6944i	− 0.708190i	−0.935210	−	0.354095i	$-0.884789\pi$
$97$	− 68.6944i	− 0.708190i	0.935210	−	0.354095i	$-0.115211\pi$
$98$	0	0
$99$	32.4659	0.327939
$100$	−10.0000	−0.100000
$101$	− 8.03902i	− 0.0795942i	−0.999208	−	0.0397971i	$-0.987329\pi$
$101$	− 8.03902i	− 0.0795942i	0.999208	−	0.0397971i	$-0.0126712\pi$
$102$	25.1654	0.246719
$103$	− 204.565i	− 1.98607i	−0.117839	−	0.993033i	$-0.537597\pi$
$103$	− 204.565i	− 1.98607i	0.117839	−	0.993033i	$-0.462403\pi$
$104$	− 54.5080i	− 0.524115i
$105$	0	0
$106$	−7.00124	−0.0660494
$107$	164.554	1.53789	0.768943	−	0.639317i	$-0.220783\pi$
$107$	164.554	1.53789	0.768943	+	0.639317i	$0.220783\pi$
$108$	10.3923i	0.0962250i
$109$	−79.0101	−0.724863	−0.362432	−	0.932010i	$-0.618053\pi$
$109$	−79.0101	−0.724863	−0.362432	+	0.932010i	$0.618053\pi$
$110$	34.2221i	0.311110i
$111$	87.2354i	0.785905i
$112$	0	0
$113$	84.5690	0.748398	0.374199	−	0.927348i	$-0.377918\pi$
$113$	84.5690	0.748398	0.374199	+	0.927348i	$0.377918\pi$
$114$	−51.1254	−0.448469
$115$	47.1242i	0.409776i
$116$	38.1776	0.329118
$117$	− 57.8144i	− 0.494140i
$118$	39.8621i	0.337815i
$119$	0	0
$120$	−10.9545	−0.0912871
$121$	−3.88480	−0.0321058
$122$	− 99.1207i	− 0.812465i
$123$	−39.4399	−0.320650
$124$	− 80.0338i	− 0.645434i
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	−101.777	−0.801393	−0.400697	−	0.916211i	$-0.631232\pi$
$127$	−101.777	−0.801393	−0.400697	+	0.916211i	$0.631232\pi$
$128$	−11.3137	−0.0883883
$129$	83.8844i	0.650266i
$130$	60.9418	0.468783
$131$	− 70.6982i	− 0.539681i	−0.962905	−	0.269840i	$-0.913029\pi$
$131$	− 70.6982i	− 0.539681i	0.962905	−	0.269840i	$-0.0869709\pi$
$132$	37.4884i	0.284003i
$133$	0	0
$134$	−27.3024	−0.203750
$135$	−11.6190	−0.0860663
$136$	29.0585i	0.213665i
$137$	−117.828	−0.860055	−0.430027	−	0.902816i	$-0.641496\pi$
$137$	−117.828	−0.860055	−0.430027	+	0.902816i	$0.641496\pi$
$138$	51.6220i	0.374072i
$139$	158.507i	1.14034i	0.821528	+	0.570168i	$0.193122\pi$
$139$	158.507i	1.14034i	−0.821528	+	0.570168i	$0.806878\pi$
$140$	0	0
$141$	−115.247	−0.817356
$142$	−69.9990	−0.492951
$143$	− 208.555i	− 1.45843i
$144$	−12.0000	−0.0833333
$145$	42.6839i	0.294372i
$146$	188.054i	1.28804i
$147$	0	0
$148$	−100.731	−0.680614
$149$	−295.896	−1.98588	−0.992940	−	0.118615i	$-0.962155\pi$
$149$	−295.896	−1.98588	−0.992940	+	0.118615i	$0.962155\pi$
$150$	− 12.2474i	− 0.0816497i
$151$	−125.296	−0.829773	−0.414886	−	0.909873i	$-0.636179\pi$
$151$	−125.296	−0.829773	−0.414886	+	0.909873i	$0.636179\pi$
$152$	− 59.0346i	− 0.388385i
$153$	30.8212i	0.201445i
$154$	0	0
$155$	89.4805	0.577293
$156$	66.7584	0.427938
$157$	5.33367i	0.0339724i	0.999856	+	0.0169862i	$0.00540714\pi$
$157$	5.33367i	0.0339724i	−0.999856	+	0.0169862i	$0.994593\pi$
$158$	127.393	0.806288
$159$	− 8.57473i	− 0.0539291i
$160$	− 12.6491i	− 0.0790569i
$161$	0	0
$162$	−12.7279	−0.0785674
$163$	239.246	1.46777	0.733883	−	0.679276i	$-0.237706\pi$
$163$	239.246	1.46777	0.733883	+	0.679276i	$0.237706\pi$
$164$	− 45.5413i	− 0.277691i
$165$	−41.9133	−0.254020
$166$	− 142.899i	− 0.860840i
$167$	− 310.440i	− 1.85892i	−0.368918	−	0.929462i	$-0.620272\pi$
$167$	− 310.440i	− 1.85892i	0.368918	−	0.929462i	$-0.379728\pi$
$168$	0	0
$169$	−202.390	−1.19757
$170$	−32.4884	−0.191108
$171$	− 62.6156i	− 0.366173i
$172$	−96.8613	−0.563147
$173$	− 90.9078i	− 0.525479i	−0.964867	−	0.262739i	$-0.915374\pi$
$173$	− 90.9078i	− 0.525479i	0.964867	−	0.262739i	$-0.0846259\pi$
$174$	46.7579i	0.268723i
$175$	0	0
$176$	−43.2879	−0.245954
$177$	−48.8209	−0.275825
$178$	56.0242i	0.314743i
$179$	−243.154	−1.35840	−0.679200	−	0.733953i	$-0.737673\pi$
$179$	−243.154	−1.35840	−0.679200	+	0.733953i	$0.737673\pi$
$180$	− 13.4164i	− 0.0745356i
$181$	245.993i	1.35907i	0.733641	+	0.679537i	$0.237819\pi$
$181$	245.993i	1.35907i	−0.733641	+	0.679537i	$0.762181\pi$
$182$	0	0
$183$	121.398	0.663375
$184$	−59.6079	−0.323956
$185$	− 112.620i	− 0.608759i
$186$	98.0209	0.526994
$187$	111.182i	0.594556i
$188$	− 133.076i	− 0.707851i
$189$	0	0
$190$	66.0027	0.347382
$191$	−41.2215	−0.215820	−0.107910	−	0.994161i	$-0.534416\pi$
$191$	−41.2215	−0.215820	−0.107910	+	0.994161i	$0.534416\pi$
$192$	− 13.8564i	− 0.0721688i
$193$	−189.745	−0.983137	−0.491568	−	0.870839i	$-0.663576\pi$
$193$	−189.745	−0.983137	−0.491568	+	0.870839i	$0.663576\pi$
$194$	97.1486i	0.500766i
$195$	74.6381i	0.382760i
$196$	0	0
$197$	−362.318	−1.83918	−0.919589	−	0.392882i	$-0.871478\pi$
$197$	−362.318	−1.83918	−0.919589	+	0.392882i	$0.871478\pi$
$198$	−45.9138	−0.231888
$199$	− 38.6170i	− 0.194056i	−0.995282	−	0.0970278i	$-0.969066\pi$
$199$	− 38.6170i	− 0.194056i	0.995282	−	0.0970278i	$-0.0309336\pi$
$200$	14.1421	0.0707107
$201$	− 33.4385i	− 0.166361i
$202$	11.3689i	0.0562816i
$203$	0	0
$204$	−35.5892	−0.174457
$205$	50.9167	0.248374
$206$	289.298i	1.40436i
$207$	−63.2238	−0.305429
$208$	77.0859i	0.370605i
$209$	− 225.875i	− 1.08074i
$210$	0	0
$211$	−136.551	−0.647163	−0.323581	−	0.946200i	$-0.604887\pi$
$211$	−136.551	−0.647163	−0.323581	+	0.946200i	$0.604887\pi$
$212$	9.90124	0.0467040
$213$	− 85.7309i	− 0.402493i
$214$	−232.714	−1.08745
$215$	− 108.294i	− 0.503694i
$216$	− 14.6969i	− 0.0680414i
$217$	0	0
$218$	111.737	0.512556
$219$	−230.318	−1.05168
$220$	− 48.3974i	− 0.219988i
$221$	197.990	0.895881
$222$	− 123.370i	− 0.555719i
$223$	− 154.949i	− 0.694839i	−0.937710	−	0.347419i	$-0.887058\pi$
$223$	− 154.949i	− 0.694839i	0.937710	−	0.347419i	$-0.112942\pi$
$224$	0	0
$225$	15.0000	0.0666667
$226$	−119.599	−0.529198
$227$	22.3538i	0.0984750i	0.998787	+	0.0492375i	$0.0156791\pi$
$227$	22.3538i	0.0984750i	−0.998787	+	0.0492375i	$0.984321\pi$
$228$	72.3023	0.317115
$229$	− 29.9188i	− 0.130650i	−0.997864	−	0.0653250i	$-0.979192\pi$
$229$	− 29.9188i	− 0.130650i	0.997864	−	0.0653250i	$-0.0208084\pi$
$230$	− 66.6437i	− 0.289755i
$231$	0	0
$232$	−53.9913	−0.232721
$233$	−253.082	−1.08619	−0.543095	−	0.839671i	$-0.682748\pi$
$233$	−253.082	−1.08619	−0.543095	+	0.839671i	$0.682748\pi$
$234$	81.7620i	0.349410i
$235$	148.783	0.633121
$236$	− 56.3736i	− 0.238871i
$237$	156.024i	0.658331i
$238$	0	0
$239$	121.009	0.506315	0.253158	−	0.967425i	$-0.418531\pi$
$239$	121.009	0.506315	0.253158	+	0.967425i	$0.418531\pi$
$240$	15.4919	0.0645497
$241$	288.392i	1.19665i	0.801255	+	0.598323i	$0.204166\pi$
$241$	288.392i	1.19665i	−0.801255	+	0.598323i	$0.795834\pi$
$242$	5.49394	0.0227022
$243$	− 15.5885i	− 0.0641500i
$244$	140.178i	0.574499i
$245$	0	0
$246$	55.7765	0.226734
$247$	−402.232	−1.62847
$248$	113.185i	0.456390i
$249$	175.015	0.702873
$250$	15.8114i	0.0632456i
$251$	− 422.260i	− 1.68231i	−0.540795	−	0.841155i	$-0.681876\pi$
$251$	− 422.260i	− 1.68231i	0.540795	−	0.841155i	$-0.318124\pi$
$252$	0	0
$253$	−228.069	−0.901457
$254$	143.934	0.566671
$255$	− 39.7899i	− 0.156039i
$256$	16.0000	0.0625000
$257$	− 74.4871i	− 0.289833i	−0.989444	−	0.144916i	$-0.953709\pi$
$257$	− 74.4871i	− 0.289833i	0.989444	−	0.144916i	$-0.0462913\pi$
$258$	− 118.630i	− 0.459808i
$259$	0	0
$260$	−86.1847	−0.331479
$261$	−57.2665	−0.219412
$262$	99.9823i	0.381612i
$263$	313.275	1.19116	0.595579	−	0.803296i	$-0.296923\pi$
$263$	313.275	1.19116	0.595579	+	0.803296i	$0.296923\pi$
$264$	− 53.0166i	− 0.200821i
$265$	11.0699i	0.0417733i
$266$	0	0
$267$	−68.6153	−0.256986
$268$	38.6115	0.144073
$269$	− 159.362i	− 0.592423i	−0.955122	−	0.296211i	$-0.904277\pi$
$269$	− 159.362i	− 0.592423i	0.955122	−	0.296211i	$-0.0957233\pi$
$270$	16.4317	0.0608581
$271$	188.264i	0.694701i	0.937735	+	0.347350i	$0.112919\pi$
$271$	188.264i	0.694701i	−0.937735	+	0.347350i	$0.887081\pi$
$272$	− 41.0949i	− 0.151084i
$273$	0	0
$274$	166.633	0.608151
$275$	54.1099	0.196763
$276$	− 73.0045i	− 0.264509i
$277$	7.04744	0.0254420	0.0127210	−	0.999919i	$-0.495951\pi$
$277$	7.04744	0.0254420	0.0127210	+	0.999919i	$0.495951\pi$
$278$	− 224.162i	− 0.806340i
$279$	120.051i	0.430289i
$280$	0	0
$281$	198.386	0.705998	0.352999	−	0.935624i	$-0.385162\pi$
$281$	198.386	0.705998	0.352999	+	0.935624i	$0.385162\pi$
$282$	162.984	0.577958
$283$	− 189.128i	− 0.668297i	−0.942520	−	0.334149i	$-0.891551\pi$
$283$	− 189.128i	− 0.668297i	0.942520	−	0.334149i	$-0.108449\pi$
$284$	98.9936	0.348569
$285$	80.8364i	0.283637i
$286$	294.942i	1.03127i
$287$	0	0
$288$	16.9706	0.0589256
$289$	183.451	0.634777
$290$	− 60.3641i	− 0.208152i
$291$	−118.982	−0.408874
$292$	− 265.949i	− 0.910783i
$293$	− 486.090i	− 1.65901i	−0.558499	−	0.829505i	$-0.688622\pi$
$293$	− 486.090i	− 1.65901i	0.558499	−	0.829505i	$-0.311378\pi$
$294$	0	0
$295$	63.0276	0.213653
$296$	142.455	0.481266
$297$	− 56.2326i	− 0.189336i
$298$	418.460	1.40423
$299$	406.138i	1.35832i
$300$	17.3205i	0.0577350i
$301$	0	0
$302$	177.195	0.586738
$303$	−13.9240	−0.0459538
$304$	83.4875i	0.274630i
$305$	−156.724	−0.513848
$306$	− 43.5877i	− 0.142443i
$307$	− 427.589i	− 1.39280i	−0.717655	−	0.696399i	$-0.754784\pi$
$307$	− 427.589i	− 1.39280i	0.717655	−	0.696399i	$-0.245216\pi$
$308$	0	0
$309$	−354.317	−1.14666
$310$	−126.545	−0.408208
$311$	− 360.095i	− 1.15786i	−0.815376	−	0.578931i	$-0.803470\pi$
$311$	− 360.095i	− 1.15786i	0.815376	−	0.578931i	$-0.196530\pi$
$312$	−94.4106	−0.302598
$313$	− 583.928i	− 1.86558i	−0.360415	−	0.932792i	$-0.617365\pi$
$313$	− 583.928i	− 1.86558i	0.360415	−	0.932792i	$-0.382635\pi$
$314$	− 7.54295i	− 0.0240221i
$315$	0	0
$316$	−180.162	−0.570132
$317$	361.400	1.14006	0.570032	−	0.821623i	$-0.306931\pi$
$317$	361.400	1.14006	0.570032	+	0.821623i	$0.306931\pi$
$318$	12.1265i	0.0381336i
$319$	−206.579	−0.647582
$320$	17.8885i	0.0559017i
$321$	− 285.016i	− 0.887899i
$322$	0	0
$323$	214.432	0.663875
$324$	18.0000	0.0555556
$325$	− 96.3574i	− 0.296484i
$326$	−338.345	−1.03787
$327$	136.849i	0.418500i
$328$	64.4051i	0.196357i
$329$	0	0
$330$	59.2744	0.179619
$331$	295.986	0.894217	0.447108	−	0.894480i	$-0.352454\pi$
$331$	295.986	0.894217	0.447108	+	0.894480i	$0.352454\pi$
$332$	202.090i	0.608706i
$333$	151.096	0.453742
$334$	439.029i	1.31446i
$335$	43.1689i	0.128863i
$336$	0	0
$337$	−22.0162	−0.0653300	−0.0326650	−	0.999466i	$-0.510399\pi$
$337$	−22.0162	−0.0653300	−0.0326650	+	0.999466i	$0.510399\pi$
$338$	286.222	0.846812
$339$	− 146.478i	− 0.432088i
$340$	45.9455	0.135134
$341$	433.062i	1.26998i
$342$	88.5519i	0.258924i
$343$	0	0
$344$	136.983	0.398205
$345$	81.6215	0.236584
$346$	128.563i	0.371570i
$347$	−283.906	−0.818173	−0.409086	−	0.912496i	$-0.634153\pi$
$347$	−283.906	−0.818173	−0.409086	+	0.912496i	$0.634153\pi$
$348$	− 66.1256i	− 0.190016i
$349$	317.175i	0.908811i	0.890795	+	0.454406i	$0.150148\pi$
$349$	317.175i	0.908811i	−0.890795	+	0.454406i	$0.849852\pi$
$350$	0	0
$351$	−100.138	−0.285292
$352$	61.2183	0.173916
$353$	− 116.942i	− 0.331282i	−0.986186	−	0.165641i	$-0.947031\pi$
$353$	− 116.942i	− 0.331282i	0.986186	−	0.165641i	$-0.0529692\pi$
$354$	69.0432	0.195037
$355$	110.678i	0.311769i
$356$	− 79.2302i	− 0.222557i
$357$	0	0
$358$	343.871	0.960534
$359$	233.586	0.650657	0.325329	−	0.945601i	$-0.394525\pi$
$359$	233.586	0.650657	0.325329	+	0.945601i	$0.394525\pi$
$360$	18.9737i	0.0527046i
$361$	−74.6351	−0.206745
$362$	− 347.886i	− 0.961011i
$363$	6.72867i	0.0185363i
$364$	0	0
$365$	297.340	0.814629
$366$	−171.682	−0.469077
$367$	512.073i	1.39530i	0.716441	+	0.697648i	$0.245770\pi$
$367$	512.073i	1.39530i	−0.716441	+	0.697648i	$0.754230\pi$
$368$	84.2984	0.229072
$369$	68.3119i	0.185127i
$370$	159.269i	0.430458i
$371$	0	0
$372$	−138.623	−0.372641
$373$	−556.407	−1.49171	−0.745854	−	0.666109i	$-0.767958\pi$
$373$	−556.407	−1.49171	−0.745854	+	0.666109i	$0.767958\pi$
$374$	− 157.235i	− 0.420415i
$375$	−19.3649	−0.0516398
$376$	188.198i	0.500526i
$377$	367.870i	0.975782i
$378$	0	0
$379$	−536.301	−1.41504	−0.707521	−	0.706692i	$-0.750187\pi$
$379$	−536.301	−1.41504	−0.707521	+	0.706692i	$0.750187\pi$
$380$	−93.3419	−0.245636
$381$	176.283i	0.462685i
$382$	58.2961	0.152608
$383$	− 29.3507i	− 0.0766336i	−0.999266	−	0.0383168i	$-0.987800\pi$
$383$	− 29.3507i	− 0.0766336i	0.999266	−	0.0383168i	$-0.0121996\pi$
$384$	19.5959i	0.0510310i
$385$	0	0
$386$	268.341	0.695183
$387$	145.292	0.375431
$388$	− 137.389i	− 0.354095i
$389$	−698.484	−1.79559	−0.897795	−	0.440414i	$-0.854832\pi$
$389$	−698.484	−1.79559	−0.897795	+	0.440414i	$0.854832\pi$
$390$	− 105.554i	− 0.270652i
$391$	− 216.514i	− 0.553745i
$392$	0	0
$393$	−122.453	−0.311585
$394$	512.395	1.30050
$395$	− 201.427i	− 0.509941i
$396$	64.9319	0.163969
$397$	204.694i	0.515602i	0.966198	+	0.257801i	$0.0829980\pi$
$397$	204.694i	0.515602i	−0.966198	+	0.257801i	$0.917002\pi$
$398$	54.6127i	0.137218i
$399$	0	0
$400$	−20.0000	−0.0500000
$401$	429.967	1.07224	0.536119	−	0.844142i	$-0.319890\pi$
$401$	429.967	1.07224	0.536119	+	0.844142i	$0.319890\pi$
$402$	47.2892i	0.117635i
$403$	771.184	1.91361
$404$	− 16.0780i	− 0.0397971i
$405$	20.1246i	0.0496904i
$406$	0	0
$407$	545.053	1.33920
$408$	50.3307	0.123360
$409$	17.7324i	0.0433554i	0.999765	+	0.0216777i	$0.00690077\pi$
$409$	17.7324i	0.0433554i	−0.999765	+	0.0216777i	$0.993099\pi$
$410$	−72.0071	−0.175627
$411$	204.083i	0.496553i
$412$	− 409.129i	− 0.993033i
$413$	0	0
$414$	89.4119	0.215971
$415$	−225.944	−0.544443
$416$	− 109.016i	− 0.262058i
$417$	274.542	0.658374
$418$	319.435i	0.764200i
$419$	440.768i	1.05195i	0.850499	+	0.525977i	$0.176300\pi$
$419$	440.768i	1.05195i	−0.850499	+	0.525977i	$0.823700\pi$
$420$	0	0
$421$	−143.012	−0.339696	−0.169848	−	0.985470i	$-0.554328\pi$
$421$	−143.012	−0.339696	−0.169848	+	0.985470i	$0.554328\pi$
$422$	193.113	0.457613
$423$	199.614i	0.471901i
$424$	−14.0025	−0.0330247
$425$	51.3686i	0.120867i
$426$	121.242i	0.284605i
$427$	0	0
$428$	329.108	0.768943
$429$	−361.229	−0.842025
$430$	153.151i	0.356166i
$431$	−783.217	−1.81721	−0.908604	−	0.417659i	$-0.862851\pi$
$431$	−783.217	−1.81721	−0.908604	+	0.417659i	$0.862851\pi$
$432$	20.7846i	0.0481125i
$433$	286.669i	0.662053i	0.943622	+	0.331026i	$0.107395\pi$
$433$	286.669i	0.662053i	−0.943622	+	0.331026i	$0.892605\pi$
$434$	0	0
$435$	73.9307	0.169956
$436$	−158.020	−0.362432
$437$	439.866i	1.00656i
$438$	325.719	0.743651
$439$	340.655i	0.775980i	0.921664	+	0.387990i	$0.126830\pi$
$439$	340.655i	0.775980i	−0.921664	+	0.387990i	$0.873170\pi$
$440$	68.4442i	0.155555i
$441$	0	0
$442$	−280.000	−0.633484
$443$	395.259	0.892232	0.446116	−	0.894975i	$-0.352807\pi$
$443$	395.259	0.892232	0.446116	+	0.894975i	$0.352807\pi$
$444$	174.471i	0.392952i
$445$	88.5820	0.199061
$446$	219.131i	0.491325i
$447$	512.507i	1.14655i
$448$	0	0
$449$	665.078	1.48124	0.740621	−	0.671923i	$-0.234531\pi$
$449$	665.078	1.48124	0.740621	+	0.671923i	$0.234531\pi$
$450$	−21.2132	−0.0471405
$451$	246.423i	0.546393i
$452$	169.138	0.374199
$453$	217.019i	0.479070i
$454$	− 31.6131i	− 0.0696323i
$455$	0	0
$456$	−102.251	−0.224234
$457$	510.937	1.11802	0.559012	−	0.829160i	$-0.311181\pi$
$457$	510.937	1.11802	0.559012	+	0.829160i	$0.311181\pi$
$458$	42.3116i	0.0923834i
$459$	53.3838	0.116305
$460$	94.2484i	0.204888i
$461$	− 174.303i	− 0.378097i	−0.981968	−	0.189049i	$-0.939460\pi$
$461$	− 174.303i	− 0.378097i	0.981968	−	0.189049i	$-0.0605404\pi$
$462$	0	0
$463$	755.187	1.63107	0.815537	−	0.578705i	$-0.196442\pi$
$463$	755.187	1.63107	0.815537	+	0.578705i	$0.196442\pi$
$464$	76.3553	0.164559
$465$	− 154.985i	− 0.333300i
$466$	357.913	0.768053
$467$	566.573i	1.21322i	0.795000	+	0.606609i	$0.207471\pi$
$467$	566.573i	1.21322i	−0.795000	+	0.606609i	$0.792529\pi$
$468$	− 115.629i	− 0.247070i
$469$	0	0
$470$	−210.412	−0.447684
$471$	9.23819	0.0196140
$472$	79.7243i	0.168907i
$473$	524.115	1.10807
$474$	− 220.652i	− 0.465510i
$475$	− 104.359i	− 0.219704i
$476$	0	0
$477$	−14.8519	−0.0311360
$478$	−171.133	−0.358019
$479$	514.172i	1.07343i	0.843764	+	0.536714i	$0.180334\pi$
$479$	514.172i	1.07343i	−0.843764	+	0.536714i	$0.819666\pi$
$480$	−21.9089	−0.0456435
$481$	− 970.616i	− 2.01791i
$482$	− 407.848i	− 0.846157i
$483$	0	0
$484$	−7.76960	−0.0160529
$485$	153.605	0.316712
$486$	22.0454i	0.0453609i
$487$	327.327	0.672129	0.336064	−	0.941839i	$-0.390904\pi$
$487$	327.327	0.672129	0.336064	+	0.941839i	$0.390904\pi$
$488$	− 198.241i	− 0.406232i
$489$	− 414.386i	− 0.847415i
$490$	0	0
$491$	−41.8889	−0.0853134	−0.0426567	−	0.999090i	$-0.513582\pi$
$491$	−41.8889	−0.0853134	−0.0426567	+	0.999090i	$0.513582\pi$
$492$	−78.8798	−0.160325
$493$	− 196.113i	− 0.397795i
$494$	568.842	1.15150
$495$	72.5960i	0.146659i
$496$	− 160.068i	− 0.322717i
$497$	0	0
$498$	−247.509	−0.497006
$499$	415.370	0.832406	0.416203	−	0.909272i	$-0.363361\pi$
$499$	415.370	0.832406	0.416203	+	0.909272i	$0.363361\pi$
$500$	− 22.3607i	− 0.0447214i
$501$	−537.698	−1.07325
$502$	597.165i	1.18957i
$503$	− 51.7604i	− 0.102903i	−0.998675	−	0.0514517i	$-0.983615\pi$
$503$	− 51.7604i	− 0.102903i	0.998675	−	0.0514517i	$-0.0163848\pi$
$504$	0	0
$505$	17.9758	0.0355956
$506$	322.538	0.637427
$507$	350.549i	0.691419i
$508$	−203.554	−0.400697
$509$	− 158.097i	− 0.310604i	−0.987867	−	0.155302i	$-0.950365\pi$
$509$	− 158.097i	− 0.310604i	0.987867	−	0.155302i	$-0.0496350\pi$
$510$	56.2715i	0.110336i
$511$	0	0
$512$	−22.6274	−0.0441942
$513$	−108.453	−0.211410
$514$	105.341i	0.204943i
$515$	457.421	0.888195
$516$	167.769i	0.325133i
$517$	720.073i	1.39279i
$518$	0	0
$519$	−157.457	−0.303385
$520$	121.884	0.234391
$521$	353.586i	0.678668i	0.940666	+	0.339334i	$0.110202\pi$
$521$	353.586i	0.678668i	−0.940666	+	0.339334i	$0.889798\pi$
$522$	80.9870	0.155147
$523$	119.551i	0.228587i	0.993447	+	0.114294i	$0.0364605\pi$
$523$	119.551i	0.228587i	−0.993447	+	0.114294i	$0.963540\pi$
$524$	− 141.396i	− 0.269840i
$525$	0	0
$526$	−443.037	−0.842277
$527$	−411.122	−0.780118
$528$	74.9769i	0.142002i
$529$	−84.8617	−0.160419
$530$	− 15.6552i	− 0.0295382i
$531$	84.5604i	0.159247i
$532$	0	0
$533$	438.824	0.823309
$534$	97.0368	0.181717
$535$	367.954i	0.687764i
$536$	−54.6049	−0.101875
$537$	421.155i	0.784273i
$538$	225.372i	0.418906i
$539$	0	0
$540$	−23.2379	−0.0430331
$541$	545.383	1.00810	0.504051	−	0.863674i	$-0.331843\pi$
$541$	545.383	1.00810	0.504051	+	0.863674i	$0.331843\pi$
$542$	− 266.245i	− 0.491228i
$543$	426.072	0.784662
$544$	58.1169i	0.106833i
$545$	− 176.672i	− 0.324169i
$546$	0	0
$547$	−117.783	−0.215325	−0.107663	−	0.994188i	$-0.534337\pi$
$547$	−117.783	−0.215325	−0.107663	+	0.994188i	$0.534337\pi$
$548$	−235.655	−0.430027
$549$	− 210.267i	− 0.383000i
$550$	−76.5229	−0.139133
$551$	398.419i	0.723084i
$552$	103.244i	0.187036i
$553$	0	0
$554$	−9.96659	−0.0179902
$555$	−195.064	−0.351467
$556$	317.014i	0.570168i
$557$	−615.488	−1.10501	−0.552503	−	0.833511i	$-0.686327\pi$
$557$	−615.488	−1.10501	−0.552503	+	0.833511i	$0.686327\pi$
$558$	− 169.777i	− 0.304260i
$559$	− 933.330i	− 1.66964i
$560$	0	0
$561$	192.573	0.343267
$562$	−280.560	−0.499216
$563$	− 479.255i	− 0.851253i	−0.904899	−	0.425626i	$-0.860054\pi$
$563$	− 479.255i	− 0.851253i	0.904899	−	0.425626i	$-0.139946\pi$
$564$	−230.494	−0.408678
$565$	189.102i	0.334694i
$566$	267.468i	0.472558i
$567$	0	0
$568$	−139.998	−0.246475
$569$	−457.347	−0.803774	−0.401887	−	0.915689i	$-0.631645\pi$
$569$	−457.347	−0.803774	−0.401887	+	0.915689i	$0.631645\pi$
$570$	− 114.320i	− 0.200561i
$571$	373.202	0.653595	0.326797	−	0.945094i	$-0.394031\pi$
$571$	373.202	0.653595	0.326797	+	0.945094i	$0.394031\pi$
$572$	− 417.111i	− 0.729215i
$573$	71.3978i	0.124604i
$574$	0	0
$575$	−105.373	−0.183257
$576$	−24.0000	−0.0416667
$577$	− 891.554i	− 1.54515i	−0.634921	−	0.772577i	$-0.718967\pi$
$577$	− 891.554i	− 1.54515i	0.634921	−	0.772577i	$-0.281033\pi$
$578$	−259.438	−0.448855
$579$	328.649i	0.567614i
$580$	85.3678i	0.147186i
$581$	0	0
$582$	168.266	0.289117
$583$	−53.5755	−0.0918962
$584$	376.108i	0.644021i
$585$	129.277	0.220986
$586$	687.435i	1.17310i
$587$	− 786.758i	− 1.34030i	−0.742224	−	0.670151i	$-0.766229\pi$
$587$	− 786.758i	− 1.34030i	0.742224	−	0.670151i	$-0.233771\pi$
$588$	0	0
$589$	835.227	1.41804
$590$	−89.1344	−0.151075
$591$	627.553i	1.06185i
$592$	−201.462	−0.340307
$593$	625.352i	1.05456i	0.849693	+	0.527278i	$0.176788\pi$
$593$	625.352i	1.05456i	−0.849693	+	0.527278i	$0.823212\pi$
$594$	79.5250i	0.133880i
$595$	0	0
$596$	−591.792	−0.992940
$597$	−66.8867	−0.112038
$598$	− 574.367i	− 0.960479i
$599$	150.495	0.251244	0.125622	−	0.992078i	$-0.459907\pi$
$599$	150.495	0.251244	0.125622	+	0.992078i	$0.459907\pi$
$600$	− 24.4949i	− 0.0408248i
$601$	521.601i	0.867888i	0.900940	+	0.433944i	$0.142878\pi$
$601$	521.601i	0.867888i	−0.900940	+	0.433944i	$0.857122\pi$
$602$	0	0
$603$	−57.9172	−0.0960485
$604$	−250.591	−0.414886
$605$	− 8.68668i	− 0.0143581i
$606$	19.6915	0.0324942
$607$	696.750i	1.14786i	0.818905	+	0.573930i	$0.194582\pi$
$607$	696.750i	1.14786i	−0.818905	+	0.573930i	$0.805418\pi$
$608$	− 118.069i	− 0.194193i
$609$	0	0
$610$	221.641	0.363345
$611$	1282.29	2.09867
$612$	61.6423i	0.100723i
$613$	54.2489	0.0884974	0.0442487	−	0.999021i	$-0.485911\pi$
$613$	54.2489	0.0884974	0.0442487	+	0.999021i	$0.485911\pi$
$614$	604.702i	0.984857i
$615$	− 88.1903i	− 0.143399i
$616$	0	0
$617$	−969.852	−1.57188	−0.785941	−	0.618301i	$-0.787821\pi$
$617$	−969.852	−1.57188	−0.785941	+	0.618301i	$0.787821\pi$
$618$	501.079	0.810808
$619$	− 128.449i	− 0.207511i	−0.994603	−	0.103756i	$-0.966914\pi$
$619$	− 128.449i	− 0.207511i	0.994603	−	0.103756i	$-0.0330860\pi$
$620$	178.961	0.288647
$621$	109.507i	0.176339i
$622$	509.252i	0.818733i
$623$	0	0
$624$	133.517	0.213969
$625$	25.0000	0.0400000
$626$	825.799i	1.31917i
$627$	−391.227	−0.623966
$628$	10.6673i	0.0169862i
$629$	517.440i	0.822639i
$630$	0	0
$631$	115.457	0.182975	0.0914877	−	0.995806i	$-0.470838\pi$
$631$	115.457	0.182975	0.0914877	+	0.995806i	$0.470838\pi$
$632$	254.787	0.403144
$633$	236.514i	0.373640i
$634$	−511.097	−0.806147
$635$	− 227.580i	− 0.358394i
$636$	− 17.1495i	− 0.0269646i
$637$	0	0
$638$	292.146	0.457910
$639$	−148.490	−0.232379
$640$	− 25.2982i	− 0.0395285i
$641$	−952.498	−1.48596	−0.742978	−	0.669316i	$-0.766587\pi$
$641$	−952.498	−1.48596	−0.742978	+	0.669316i	$0.766587\pi$
$642$	403.073i	0.627840i
$643$	− 253.254i	− 0.393863i	−0.980417	−	0.196931i	$-0.936902\pi$
$643$	− 253.254i	− 0.393863i	0.980417	−	0.196931i	$-0.0630976\pi$
$644$	0	0
$645$	−187.571	−0.290808
$646$	−303.252	−0.469431
$647$	1004.78i	1.55298i	0.630132	+	0.776488i	$0.283001\pi$
$647$	1004.78i	1.55298i	−0.630132	+	0.776488i	$0.716999\pi$
$648$	−25.4558	−0.0392837
$649$	305.037i	0.470010i
$650$	136.270i	0.209646i
$651$	0	0
$652$	478.492	0.733883
$653$	1065.59	1.63184	0.815922	−	0.578162i	$-0.196230\pi$
$653$	1065.59	1.63184	0.815922	+	0.578162i	$0.196230\pi$
$654$	− 193.534i	− 0.295924i
$655$	158.086	0.241353
$656$	− 91.0826i	− 0.138845i
$657$	398.923i	0.607189i
$658$	0	0
$659$	432.265	0.655941	0.327970	−	0.944688i	$-0.393635\pi$
$659$	432.265	0.655941	0.327970	+	0.944688i	$0.393635\pi$
$660$	−83.8267	−0.127010
$661$	378.310i	0.572329i	0.958180	+	0.286165i	$0.0923804\pi$
$661$	378.310i	0.572329i	−0.958180	+	0.286165i	$0.907620\pi$
$662$	−418.587	−0.632307
$663$	− 342.928i	− 0.517237i
$664$	− 285.799i	− 0.430420i
$665$	0	0
$666$	−213.682	−0.320844
$667$	402.289	0.603132
$668$	− 620.880i	− 0.929462i
$669$	−268.380	−0.401165
$670$	− 61.0501i	− 0.0911196i
$671$	− 758.501i	− 1.13040i
$672$	0	0
$673$	−689.666	−1.02476	−0.512382	−	0.858758i	$-0.671237\pi$
$673$	−689.666	−1.02476	−0.512382	+	0.858758i	$0.671237\pi$
$674$	31.1356	0.0461953
$675$	− 25.9808i	− 0.0384900i
$676$	−404.780	−0.598786
$677$	− 379.508i	− 0.560572i	−0.959916	−	0.280286i	$-0.909571\pi$
$677$	− 379.508i	− 0.560572i	0.959916	−	0.280286i	$-0.0904294\pi$
$678$	207.151i	0.305532i
$679$	0	0
$680$	−64.9767	−0.0955540
$681$	38.7179	0.0568545
$682$	− 612.442i	− 0.898009i
$683$	747.507	1.09445	0.547223	−	0.836987i	$-0.315685\pi$
$683$	747.507	1.09445	0.547223	+	0.836987i	$0.315685\pi$
$684$	− 125.231i	− 0.183087i
$685$	− 263.470i	− 0.384628i
$686$	0	0
$687$	−51.8209	−0.0754308
$688$	−193.723	−0.281574
$689$	95.4058i	0.138470i
$690$	−115.430	−0.167290
$691$	− 890.304i	− 1.28843i	−0.764845	−	0.644214i	$-0.777185\pi$
$691$	− 890.304i	− 1.28843i	0.764845	−	0.644214i	$-0.222815\pi$
$692$	− 181.816i	− 0.262739i
$693$	0	0
$694$	401.504	0.578536
$695$	−354.432	−0.509974
$696$	93.5157i	0.134362i
$697$	−233.939	−0.335637
$698$	− 448.553i	− 0.642627i
$699$	438.352i	0.627112i
$700$	0	0
$701$	−650.703	−0.928250	−0.464125	−	0.885770i	$-0.653631\pi$
$701$	−650.703	−0.928250	−0.464125	+	0.885770i	$0.653631\pi$
$702$	141.616	0.201732
$703$	− 1051.22i	− 1.49533i
$704$	−86.5758	−0.122977
$705$	− 257.700i	− 0.365533i
$706$	165.382i	0.234251i
$707$	0	0
$708$	−97.6419	−0.137912
$709$	−392.855	−0.554097	−0.277048	−	0.960856i	$-0.589356\pi$
$709$	−392.855	−0.554097	−0.277048	+	0.960856i	$0.589356\pi$
$710$	− 156.523i	− 0.220454i
$711$	270.242	0.380088
$712$	112.048i	0.157371i
$713$	− 843.339i	− 1.18280i
$714$	0	0
$715$	466.344	0.652230
$716$	−486.307	−0.679200
$717$	− 209.594i	− 0.292321i
$718$	−330.340	−0.460084
$719$	− 1039.14i	− 1.44525i	−0.691238	−	0.722627i	$-0.742935\pi$
$719$	− 1039.14i	− 1.44525i	0.691238	−	0.722627i	$-0.257065\pi$
$720$	− 26.8328i	− 0.0372678i
$721$	0	0
$722$	105.550	0.146191
$723$	499.509	0.690884
$724$	491.985i	0.679537i
$725$	−95.4441	−0.131647
$726$	− 9.51578i	− 0.0131071i
$727$	610.568i	0.839846i	0.907560	+	0.419923i	$0.137943\pi$
$727$	610.568i	0.839846i	−0.907560	+	0.419923i	$0.862057\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	−420.502	−0.576030
$731$	497.563i	0.680661i
$732$	242.795	0.331687
$733$	− 1079.15i	− 1.47224i	−0.676850	−	0.736121i	$-0.736656\pi$
$733$	− 1079.15i	− 1.47224i	0.676850	−	0.736121i	$-0.263344\pi$
$734$	− 724.181i	− 0.986623i
$735$	0	0
$736$	−119.216	−0.161978
$737$	−208.926	−0.283482
$738$	− 96.6077i	− 0.130905i
$739$	756.164	1.02323	0.511613	−	0.859216i	$-0.329048\pi$
$739$	756.164	1.02323	0.511613	+	0.859216i	$0.329048\pi$
$740$	− 225.241i	− 0.304380i
$741$	696.686i	0.940197i
$742$	0	0
$743$	963.993	1.29743	0.648717	−	0.761030i	$-0.275306\pi$
$743$	963.993	1.29743	0.648717	+	0.761030i	$0.275306\pi$
$744$	196.042	0.263497
$745$	− 661.644i	− 0.888113i
$746$	786.879	1.05480
$747$	− 303.135i	− 0.405804i
$748$	222.364i	0.297278i
$749$	0	0
$750$	27.3861	0.0365148
$751$	833.612	1.11000	0.555001	−	0.831850i	$-0.312718\pi$
$751$	833.612	1.11000	0.555001	+	0.831850i	$0.312718\pi$
$752$	− 266.152i	− 0.353925i
$753$	−731.375	−0.971282
$754$	− 520.246i	− 0.689982i
$755$	− 280.170i	− 0.371086i
$756$	0	0
$757$	−744.966	−0.984103	−0.492051	−	0.870566i	$-0.663753\pi$
$757$	−744.966	−0.984103	−0.492051	+	0.870566i	$0.663753\pi$
$758$	758.444	1.00059
$759$	395.027i	0.520457i
$760$	132.005	0.173691
$761$	74.0761i	0.0973405i	0.998815	+	0.0486702i	$0.0154983\pi$
$761$	74.0761i	0.0973405i	−0.998815	+	0.0486702i	$0.984502\pi$
$762$	− 249.302i	− 0.327167i
$763$	0	0
$764$	−82.4431	−0.107910
$765$	−68.9182	−0.0900892
$766$	41.5081i	0.0541881i
$767$	543.201	0.708215
$768$	− 27.7128i	− 0.0360844i
$769$	− 961.553i	− 1.25039i	−0.780467	−	0.625197i	$-0.785019\pi$
$769$	− 961.553i	− 1.25039i	0.780467	−	0.625197i	$-0.214981\pi$
$770$	0	0
$771$	−129.015	−0.167335
$772$	−379.491	−0.491568
$773$	− 1375.45i	− 1.77936i	−0.456584	−	0.889680i	$-0.650927\pi$
$773$	− 1375.45i	− 1.77936i	0.456584	−	0.889680i	$-0.349073\pi$
$774$	−205.474	−0.265470
$775$	200.084i	0.258173i
$776$	194.297i	0.250383i
$777$	0	0
$778$	987.806	1.26967
$779$	475.266	0.610098
$780$	149.276i	0.191380i
$781$	−535.653	−0.685855
$782$	306.198i	0.391557i
$783$	99.1884i	0.126677i
$784$	0	0
$785$	−11.9264	−0.0151929
$786$	173.174	0.220324
$787$	380.681i	0.483711i	0.970312	+	0.241856i	$0.0777560\pi$
$787$	380.681i	0.483711i	−0.970312	+	0.241856i	$0.922244\pi$
$788$	−724.636	−0.919589
$789$	− 542.608i	− 0.687716i
$790$	284.860i	0.360583i
$791$	0	0
$792$	−91.8275	−0.115944
$793$	−1350.72	−1.70330
$794$	− 289.481i	− 0.364586i
$795$	19.1737	0.0241178
$796$	− 77.2341i	− 0.0970278i
$797$	511.440i	0.641706i	0.947129	+	0.320853i	$0.103970\pi$
$797$	511.440i	0.641706i	−0.947129	+	0.320853i	$0.896030\pi$
$798$	0	0
$799$	−683.593	−0.855560
$800$	28.2843	0.0353553
$801$	118.845i	0.148371i
$802$	−608.066	−0.758187
$803$	1439.04i	1.79209i
$804$	− 66.8770i	− 0.0831804i
$805$	0	0
$806$	−1090.62	−1.35313
$807$	−276.023	−0.342036
$808$	22.7378i	0.0281408i
$809$	1158.37	1.43186	0.715930	−	0.698173i	$-0.246003\pi$
$809$	1158.37	1.43186	0.715930	+	0.698173i	$0.246003\pi$
$810$	− 28.4605i	− 0.0351364i
$811$	− 92.0692i	− 0.113526i	−0.998388	−	0.0567628i	$-0.981922\pi$
$811$	− 92.0692i	− 0.113526i	0.998388	−	0.0567628i	$-0.0180779\pi$
$812$	0	0
$813$	326.083	0.401086
$814$	−770.822	−0.946956
$815$	534.970i	0.656405i
$816$	−71.1784	−0.0872285
$817$	− 1010.84i	− 1.23726i
$818$	− 25.0774i	− 0.0306569i
$819$	0	0
$820$	101.833	0.124187
$821$	−986.935	−1.20211	−0.601057	−	0.799206i	$-0.705253\pi$
$821$	−986.935	−1.20211	−0.601057	+	0.799206i	$0.705253\pi$
$822$	− 288.617i	− 0.351116i
$823$	555.497	0.674966	0.337483	−	0.941332i	$-0.390424\pi$
$823$	555.497	0.674966	0.337483	+	0.941332i	$0.390424\pi$
$824$	578.596i	0.702180i
$825$	− 93.7211i	− 0.113601i
$826$	0	0
$827$	−1323.46	−1.60032	−0.800160	−	0.599787i	$-0.795252\pi$
$827$	−1323.46	−1.60032	−0.800160	+	0.599787i	$0.795252\pi$
$828$	−126.448	−0.152714
$829$	1052.98i	1.27018i	0.772440	+	0.635088i	$0.219036\pi$
$829$	1052.98i	1.27018i	−0.772440	+	0.635088i	$0.780964\pi$
$830$	319.533	0.384979
$831$	− 12.2065i	− 0.0146890i
$832$	154.172i	0.185303i
$833$	0	0
$834$	−388.261	−0.465540
$835$	694.165	0.831336
$836$	− 451.750i	− 0.540371i
$837$	207.934	0.248428
$838$	− 623.341i	− 0.743843i
$839$	1254.98i	1.49580i	0.663810	+	0.747901i	$0.268938\pi$
$839$	1254.98i	1.49580i	−0.663810	+	0.747901i	$0.731062\pi$
$840$	0	0
$841$	−476.617	−0.566727
$842$	202.249	0.240201
$843$	− 343.614i	− 0.407608i
$844$	−273.103	−0.323581
$845$	− 452.557i	− 0.535571i
$846$	− 282.297i	− 0.333684i
$847$	0	0
$848$	19.8025	0.0233520
$849$	−327.580	−0.385842
$850$	− 72.6462i	− 0.0854661i
$851$	−1061.43	−1.24727
$852$	− 171.462i	− 0.201246i
$853$	− 454.825i	− 0.533207i	−0.963806	−	0.266603i	$-0.914099\pi$
$853$	− 454.825i	− 0.533207i	0.963806	−	0.266603i	$-0.0859014\pi$
$854$	0	0
$855$	140.013	0.163758
$856$	−465.429	−0.543725
$857$	− 168.738i	− 0.196894i	−0.995142	−	0.0984470i	$-0.968613\pi$
$857$	− 168.738i	− 0.196894i	0.995142	−	0.0984470i	$-0.0313875\pi$
$858$	510.855	0.595402
$859$	817.199i	0.951337i	0.879625	+	0.475669i	$0.157794\pi$
$859$	817.199i	0.951337i	−0.879625	+	0.475669i	$0.842206\pi$
$860$	− 216.588i	− 0.251847i
$861$	0	0
$862$	1107.64	1.28496
$863$	−1233.90	−1.42978	−0.714890	−	0.699237i	$-0.753523\pi$
$863$	−1233.90	−1.42978	−0.714890	+	0.699237i	$0.753523\pi$
$864$	− 29.3939i	− 0.0340207i
$865$	203.276	0.235001
$866$	− 405.411i	− 0.468142i
$867$	− 317.746i	− 0.366489i
$868$	0	0
$869$	974.852	1.12181
$870$	−104.554	−0.120177
$871$	372.050i	0.427153i
$872$	223.474	0.256278
$873$	206.083i	0.236063i
$874$	− 622.065i	− 0.711744i
$875$	0	0
$876$	−460.637	−0.525841
$877$	74.9557	0.0854683	0.0427342	−	0.999086i	$-0.486393\pi$
$877$	74.9557	0.0854683	0.0427342	+	0.999086i	$0.486393\pi$
$878$	− 481.759i	− 0.548701i
$879$	−841.933	−0.957830
$880$	− 96.7947i	− 0.109994i
$881$	461.343i	0.523658i	0.965114	+	0.261829i	$0.0843257\pi$
$881$	461.343i	0.523658i	−0.965114	+	0.261829i	$0.915674\pi$
$882$	0	0
$883$	237.840	0.269354	0.134677	−	0.990890i	$-0.457000\pi$
$883$	237.840	0.269354	0.134677	+	0.990890i	$0.457000\pi$
$884$	395.980	0.447941
$885$	− 109.167i	− 0.123352i
$886$	−558.981	−0.630904
$887$	− 9.11359i	− 0.0102746i	−0.999987	−	0.00513731i	$-0.998365\pi$
$887$	− 9.11359i	− 0.0102746i	0.999987	−	0.00513731i	$-0.00163526\pi$
$888$	− 246.739i	− 0.277859i
$889$	0	0
$890$	−125.274	−0.140757
$891$	−97.3978	−0.109313
$892$	− 309.898i	− 0.347419i
$893$	1388.77	1.55518
$894$	− 724.795i	− 0.810732i
$895$	− 543.708i	− 0.607495i
$896$	0	0
$897$	703.452	0.784228
$898$	−940.562	−1.04740
$899$	− 763.875i	− 0.849694i
$900$	30.0000	0.0333333
$901$	− 50.8613i	− 0.0564498i
$902$	− 348.495i	− 0.386358i
$903$	0	0
$904$	−239.197	−0.264599
$905$	−550.056	−0.607797
$906$	− 306.911i	− 0.338753i
$907$	840.703	0.926905	0.463453	−	0.886122i	$-0.346610\pi$
$907$	840.703	0.926905	0.463453	+	0.886122i	$0.346610\pi$
$908$	44.7076i	0.0492375i
$909$	24.1171i	0.0265314i
$910$	0	0
$911$	−35.4735	−0.0389390	−0.0194695	−	0.999810i	$-0.506198\pi$
$911$	−35.4735	−0.0389390	−0.0194695	+	0.999810i	$0.506198\pi$
$912$	144.605	0.158558
$913$	− 1093.51i	− 1.19771i
$914$	−722.573	−0.790562
$915$	271.453i	0.296670i
$916$	− 59.8377i	− 0.0653250i
$917$	0	0
$918$	−75.4961	−0.0822398
$919$	431.360	0.469380	0.234690	−	0.972070i	$-0.424593\pi$
$919$	431.360	0.469380	0.234690	+	0.972070i	$0.424593\pi$
$920$	− 133.287i	− 0.144878i
$921$	−740.606	−0.804132
$922$	246.502i	0.267355i
$923$	953.876i	1.03345i
$924$	0	0
$925$	251.827	0.272245
$926$	−1068.00	−1.15334
$927$	613.694i	0.662022i
$928$	−107.983	−0.116361
$929$	− 613.207i	− 0.660072i	−0.943968	−	0.330036i	$-0.892939\pi$
$929$	− 613.207i	− 0.660072i	0.943968	−	0.330036i	$-0.107061\pi$
$930$	219.182i	0.235679i
$931$	0	0
$932$	−506.165	−0.543095
$933$	−623.703	−0.668492
$934$	− 801.255i	− 0.857875i
$935$	−248.610	−0.265894
$936$	163.524i	0.174705i
$937$	1360.68i	1.45216i	0.687609	+	0.726081i	$0.258660\pi$
$937$	1360.68i	1.45216i	−0.687609	+	0.726081i	$0.741340\pi$
$938$	0	0
$939$	−1011.39	−1.07710
$940$	297.567	0.316561
$941$	− 169.361i	− 0.179980i	−0.995943	−	0.0899900i	$-0.971316\pi$
$941$	− 169.361i	− 0.179980i	0.995943	−	0.0899900i	$-0.0286835\pi$
$942$	−13.0648	−0.0138692
$943$	− 479.882i	− 0.508889i
$944$	− 112.747i	− 0.119436i
$945$	0	0
$946$	−741.211	−0.783521
$947$	−1279.17	−1.35076	−0.675378	−	0.737472i	$-0.736019\pi$
$947$	−1279.17	−1.35076	−0.675378	+	0.737472i	$0.736019\pi$
$948$	312.049i	0.329166i
$949$	2562.61	2.70033
$950$	147.586i	0.155354i
$951$	− 625.964i	− 0.658216i
$952$	0	0
$953$	−1.43779	−0.00150870	−0.000754349	−	1.00000i	$-0.500240\pi$
$953$	−1.43779	−0.00150870	−0.000754349		1.00000i	$0.500240\pi$
$954$	21.0037	0.0220165
$955$	− 92.1742i	− 0.0965175i
$956$	242.019	0.253158
$957$	357.805i	0.373882i
$958$	− 727.149i	− 0.759028i
$959$	0	0
$960$	30.9839	0.0322749
$961$	−640.351	−0.666338
$962$	1372.66i	1.42688i
$963$	−493.662	−0.512629
$964$	576.784i	0.598323i
$965$	− 424.284i	− 0.439672i
$966$	0	0
$967$	486.815	0.503428	0.251714	−	0.967802i	$-0.419006\pi$
$967$	486.815	0.503428	0.251714	+	0.967802i	$0.419006\pi$
$968$	10.9879	0.0113511
$969$	− 371.407i	− 0.383289i
$970$	−217.231	−0.223949
$971$	− 436.725i	− 0.449768i	−0.974386	−	0.224884i	$-0.927800\pi$
$971$	− 436.725i	− 0.449768i	0.974386	−	0.224884i	$-0.0722004\pi$
$972$	− 31.1769i	− 0.0320750i
$973$	0	0
$974$	−462.910	−0.475267
$975$	−166.896	−0.171175
$976$	280.356i	0.287250i
$977$	255.034	0.261038	0.130519	−	0.991446i	$-0.458336\pi$
$977$	255.034	0.261038	0.130519	+	0.991446i	$0.458336\pi$
$978$	586.030i	0.599213i
$979$	428.714i	0.437910i
$980$	0	0
$981$	237.030	0.241621
$982$	59.2398	0.0603257
$983$	1380.89i	1.40477i	0.711799	+	0.702383i	$0.247881\pi$
$983$	1380.89i	1.40477i	−0.711799	+	0.702383i	$0.752119\pi$
$984$	111.553	0.113367
$985$	− 810.168i	− 0.822505i
$986$	277.346i	0.281284i
$987$	0	0
$988$	−804.464	−0.814234
$989$	−1020.66	−1.03201
$990$	− 102.666i	− 0.103703i
$991$	−623.108	−0.628767	−0.314384	−	0.949296i	$-0.601798\pi$
$991$	−623.108	−0.628767	−0.314384	+	0.949296i	$0.601798\pi$
$992$	226.370i	0.228195i
$993$	− 512.662i	− 0.516276i
$994$	0	0
$995$	86.3503	0.0867843
$996$	350.031	0.351436
$997$	560.317i	0.562003i	0.959707	+	0.281001i	$0.0906665\pi$
$997$	560.317i	0.562003i	−0.959707	+	0.281001i	$0.909333\pi$
$998$	−587.423	−0.588600
$999$	− 261.706i	− 0.261968i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1470.3.f.d.391.3		16
7.4	even	3		210.3.o.b.61.5	yes	16
7.5	odd	6		210.3.o.b.31.5	✓	16
7.6	odd	2	inner	1470.3.f.d.391.5		16
21.5	even	6		630.3.v.c.451.3		16
21.11	odd	6		630.3.v.c.271.3		16
35.4	even	6		1050.3.p.i.901.3		16
35.12	even	12		1050.3.q.e.199.7		32
35.18	odd	12		1050.3.q.e.649.7		32
35.19	odd	6		1050.3.p.i.451.3		16
35.32	odd	12		1050.3.q.e.649.16		32
35.33	even	12		1050.3.q.e.199.16		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.3.o.b.31.5	✓	16	7.5	odd	6
210.3.o.b.61.5	yes	16	7.4	even	3
630.3.v.c.271.3		16	21.11	odd	6
630.3.v.c.451.3		16	21.5	even	6
1050.3.p.i.451.3		16	35.19	odd	6
1050.3.p.i.901.3		16	35.4	even	6
1050.3.q.e.199.7		32	35.12	even	12
1050.3.q.e.199.16		32	35.33	even	12
1050.3.q.e.649.7		32	35.18	odd	12
1050.3.q.e.649.16		32	35.32	odd	12
1470.3.f.d.391.3		16	1.1	even	1	trivial
1470.3.f.d.391.5		16	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1470.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.73205i q^{3} +2.00000 q^{4} +2.23607i q^{5} +2.44949i q^{6} -2.82843 q^{8} -3.00000 q^{9} -3.16228i q^{10} -10.8220 q^{11} -3.46410i q^{12} +19.2715i q^{13} +3.87298 q^{15} +4.00000 q^{16} -10.2737i q^{17} +4.24264 q^{18} +20.8719i q^{19} +4.47214i q^{20} +15.3046 q^{22} +21.0746 q^{23} +4.89898i q^{24} -5.00000 q^{25} -27.2540i q^{26} +5.19615i q^{27} +19.0888 q^{29} -5.47723 q^{30} -40.0169i q^{31} -5.65685 q^{32} +18.7442i q^{33} +14.5292i q^{34} -6.00000 q^{36} -50.3654 q^{37} -29.5173i q^{38} +33.3792 q^{39} -6.32456i q^{40} -22.7706i q^{41} -48.4307 q^{43} -21.6440 q^{44} -6.70820i q^{45} -29.8040 q^{46} -66.5380i q^{47} -6.92820i q^{48} +7.07107 q^{50} -17.7946 q^{51} +38.5430i q^{52} +4.95062 q^{53} -7.34847i q^{54} -24.1987i q^{55} +36.1511 q^{57} -26.9957 q^{58} -28.1868i q^{59} +7.74597 q^{60} +70.0889i q^{61} +56.5924i q^{62} +8.00000 q^{64} -43.0923 q^{65} -26.5083i q^{66} +19.3057 q^{67} -20.5474i q^{68} -36.5023i q^{69} +49.4968 q^{71} +8.48528 q^{72} -132.974i q^{73} +71.2274 q^{74} +8.66025i q^{75} +41.7437i q^{76} -47.2053 q^{78} -90.0808 q^{79} +8.94427i q^{80} +9.00000 q^{81} +32.2026i q^{82} +101.045i q^{83} +22.9727 q^{85} +68.4913 q^{86} -33.0628i q^{87} +30.6092 q^{88} -39.6151i q^{89} +9.48683i q^{90} +42.1492 q^{92} -69.3113 q^{93} +94.0989i q^{94} -46.6709 q^{95} +9.79796i q^{96} -68.6944i q^{97} +32.4659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4} - 48 q^{9}+O(q^{10}) \) 16 * q + 32 * q^4 - 48 * q^9	\( 16 q + 32 q^{4} - 48 q^{9} + 8 q^{11} + 64 q^{16} - 48 q^{22} + 24 q^{23} - 80 q^{25} + 72 q^{29} - 96 q^{36} - 88 q^{37} - 72 q^{39} - 56 q^{43} + 16 q^{44} - 16 q^{46} + 24 q^{51} - 64 q^{53} + 144 q^{57} + 176 q^{58} + 128 q^{64} - 40 q^{65} + 328 q^{67} - 136 q^{71} + 224 q^{74} - 560 q^{79} + 144 q^{81} + 120 q^{85} + 176 q^{86} - 96 q^{88} + 48 q^{92} + 240 q^{93} - 400 q^{95} - 24 q^{99}+O(q^{100}) \) 16 * q + 32 * q^4 - 48 * q^9 + 8 * q^11 + 64 * q^16 - 48 * q^22 + 24 * q^23 - 80 * q^25 + 72 * q^29 - 96 * q^36 - 88 * q^37 - 72 * q^39 - 56 * q^43 + 16 * q^44 - 16 * q^46 + 24 * q^51 - 64 * q^53 + 144 * q^57 + 176 * q^58 + 128 * q^64 - 40 * q^65 + 328 * q^67 - 136 * q^71 + 224 * q^74 - 560 * q^79 + 144 * q^81 + 120 * q^85 + 176 * q^86 - 96 * q^88 + 48 * q^92 + 240 * q^93 - 400 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more